p adic singular integrals and their commutators in generalized Morrey spaces

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R E S E A R C H

Open Access

p

-adic singular integrals and their

commutators in generalized Morrey spaces

Hui X. Mo

1*

_{, Zhe Han}

1

_{, Liu Yang}

1

_{and Xiao J. Wang}

1

*_{Correspondence:}

[email protected]

1_{School of Science, Beijing}

University of Posts and

Telecommunications, Beijing, China

Abstract

For a prime numberp, letQpbe the ﬁeld ofp-adic numbers. In this paper, we

establish the boundedness of a class ofp-adic singular integral operators on the p-adic generalized Morrey spaces. We also consider the corresponding boundedness for the commutators generalized by thep-adic singular integral operators andp-adic Lipschitz functions orp-adic generalized Campanato functions.

MSC: 42B20; 42B25

Keywords: p-adic ﬁeld;p-adic singular integral operator; Commutator;p-adic

generalized Morrey function;p-adic generalized Campanato function;p-adic Lipschitz

function

1 Introduction

Letpbe a prime number, and letx∈Q. Then the non-Archimedeanp-adic norm|x|p

is deﬁned as follows: ifx= 0, then|0|p= 0; ifx= 0 is an arbitrary rational number with unique representationx=pγm

n, wherem,nare not divisible byp, andγ =γ(x)∈Z, then |x|p=p–γ_{. This norm has the following properties:}_|_xy_|p₌_|_x_|p|_y_|p_,_|_x₊_y_|p_≤_max_{|_x_|p_,_|_y_|p}_,

and|x|p= 0 if and only ifx= 0. Moreover, when|x|p=|y|p, we have|x+y|p=max{|x|p,|y|p}. LetQpbe the field ofp-adic numbers defined as the completion of the field of rational

numbersQwith respect to the non-Archimedeanp-adic norm| · |p. Forγ∈Z, we denote the ballBγ(a) with center ata∈Qpand radiuspγ and its boundarySγ(a) by

Bγ(a) =

x∈Qp:|x–a|p≤pγ

, Sγ(a) =

x∈Qp:|x–a|p=pγ

,

respectively. It is easy to see that

Bγ(a) =

k≤γ

Sk(a).

Forn∈N, the spaceQn

p=Qp× · · · ×Qp consists of all pointsx= (x1, . . . ,xn) where xi∈Qp,i= 1, . . . ,n,n≥1. Thep-adic norm ofQnpis deﬁned by

|x|p=max

1≤i≤n|xi|p, x∈Q n p.

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Thus it is easy to see that|x|pis a non-Archimedean norm onQn

p. The ballsBγ(a) and the

sphereSγ(a) inQn_pforγ ∈Zare deﬁned similarly to the casen= 1.

SinceQn_pis a locally compact commutative group under addition, by the standard anal-ysis there exists the Haar measuredxon the additive groupQn_p normalized by_B

0dx=

|B0|H= 1, where|E|H denotes the Haar measure of a measurable setE⊂Qn_p. Then by a simple calculation the Haar measures of any balls and spheres can be obtained. From the integral theory it is easy to see that |Bγ(a)|H=pnγ and|Sγ(a)|H=pnγ(1 –p–n) for any

a∈Qn

p. For a more complete introduction to thep-adic analysis, we refer to [1–8] and the

references therein.

Thep-adic numbers have been applied in the string theory, turbulence theory, statistical mechanics, quantum mechanics, and so forth (see [1,9,10] for detail). In the past few years, there is an increasing interest in the study of harmonic analysis onp-adic ﬁeld (see [5–8] for detail).

LetΩ∈L∞(Q_pn) be such thatΩ(pjx) =Ω(x) for allj∈Zand_|_x_|_p=1Ω(x)dx= 0. Then thep-adic singular integral operators deﬁned by Taibleson [5] are as follows:

Tk(f)(x) =

|y|p>pk

f(x–y)Ω(y)

|y|n p

dz fork∈Z.

Thep-adic singular integral operatorTis deﬁned as the limit ofTkaskgoes to –∞.

Moreover, let−→b = (b1,b2, . . . ,bm), wherebi∈Lloc(Qnp) for 1≤i≤m. Then the higher

commutator generated bybandTkcan be deﬁned as

T_kbf(x) =

|y|p>pk m

i=1

bi(x) –bi(x–y)f(x–y)

Ω(y)

|y|n p

dz fork∈Z,

and the commutator generated by −→b = (b1,b2, . . . ,bm) and thep-adic singular integral

operatorTis deﬁned as the limit ofTb

k askgoes to –∞.

Under some conditions, the authors in [5,11] showed thatTk were of type (q,q) for

1 <q<∞and of weak type (1, 1) on local ﬁelds. Wu et al. [12] established the boundedness of Tk onp-adic central Morrey spaces. Furthermore, theλ-central BMO estimates for

commutators of these singular integral operators onp-adic central Morrey spaces were obtained in [12]. Moreover, in thep-adic linear spaceQn

p, Volosivets [13] gave suﬃcient

conditions for the boundedness of the maximal function and Riesz potential inp-adic generalized Morrey spaces. Mo et al. [14] established the boundedness of the commutators generated by thep-adic Riesz potential andp-adic generalized Campanato functions inp -adic generalized Morrey spaces.

Motivated by the works of [12–14], we consider the boundedness ofTkon thep-adic

generalized Morrey type spaces, as well as the boundedness of the commutators generated byTkandp-adic generalized Campanato functions.

Throughout this paper, the letterCwill be used to denote constants varying from line to line. The relationABmeans thatA≤CBwith some positive constantCindependent of appropriate quantities.

2 Some notation and lemmas

Deﬁnition 2.1([13]) Let 1≤q<∞, and letω(x) be a nonnegative measurable function in Qn

p. A functionf ∈L q

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if

fGMq,ω= sup

a∈Qn_p,_γ_∈_Z

1

ω(Bγ(a))

1

|Bγ(a)|H

Bγ(a)

f(y)qdy

1/q

<∞,

whereω(Bγ(a)) =

Bγ(a)ω(x)dx.

Letλ∈R. Ifω(Bγ(a)) =|Bγ(a)|λ, thenGMq,ω(Qnp) is the classical Morrey spaceMq,λ(Qnp).

About the generalized Morrey space, see [15], and for the classical Morrey spaces, see [16] and so on.

Moreover, letλ∈Rand 1≤q<∞. Thep-adic central Morrey spaceCMq,λ(Qn_p) (see

[8]) is deﬁned by

fCMq,λ=sup γ∈Z

1

|Bγ(0)|1+ λq H

Bγ(0)

f(y)qdy

1/q

<∞.

Deﬁnition 2.2([17]) For 0 <β< 1, the thep-adic Lipschitz spaceΛβ(Qn_p) is deﬁned as

the set of all functionsf:Qn

p→Csuch that

fΛβ(Qnp)= sup x,h∈Qn

p,h=0

|f(x+h) –f(x)|

|h|βp

<∞.

Deﬁnition 2.3 ([13]) LetBbe a ball inQn_p, 1≤q<∞, and letω(x) be a nonnegative measurable function inQn_p. A functionf ∈Lq_loc(Qn_p) is said to belong to the generalized Campanato spaceGCq,ω(Qnp) if

fGCq,ω= sup

a∈Qnp,γ∈Z

1

ω(Bγ(a))

1

|Bγ(a)|H

Bγ(a)

f(y) –fBγ(a)

q dy

1/q

<∞,

wherefBγ(a)=

1

|Bγ(a)|H

Bγ(a)f(x)dxandω(Bγ(a)) =

Bγ(a)ω(x)dx.

The classical Campanato spaces can be found in [18, 19], and so on. The important particular case ofGCq,ω(Qn_p) isBMOq,λ(Qn_p), where 1 <q<∞and 0 <λ< 1/n. The central

BMO spaceCBMOq,λ(Qnp) is deﬁned by

f_CBMOq,λ₍_Qn_p)=sup

γ∈Z

1

|Bγ(0)|λ_H

1

|Bγ(0)|H

Bγ(0)

f(y) –fBγ(0)

q dy

1/q

<∞. (2.1)

Lemma 2.1([14]) Let1≤q<∞, and letωbe a nonnegative measurable function.Let b∈GCq,ω(Qnp).Then

|bBk(a)–bBj(a)| ≤ bGCq,ω|j–k|max

ωBk(a) ,ω

Bj(a)

for j,k∈Zand any ﬁxed a∈Qn p.

Thus, forj>k, from Lemma2.1we deduce that

Bj(a)

_b(y) –bBk(a) q

dy

1/q

≤(j+ 1 –k)Bj(a) 1/q H ω

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Lemma 2.2 ([5]) Let Ω ∈ L∞(Qn

p) be such that Ω(pjx) = Ω(x) for all j ∈ Z and

|x|p=1Ω(x)dx= 0.If

sup

|y|p=1

∞

j=1

|x|p=1

Ωx+pjy –Ω(x)dx<∞,

then for1 <p<∞,there is a constant C> 0such that

Tk(f)_Lp₍_Qn_p)≤CfLp₍_Qn_p)

for k∈Z,where C is independent of f and k∈Z.

Furthermore,T(f) =limk→–∞Tk(f) exists in theLpnorm, and

T(f)_Lp₍_Q_p)n ≤CfLp(Qnp).

Moreover, on thep-adic ﬁeld, the Riesz potentialIαpis deﬁned by

I_pαf(x) = 1

Γn(α)

Qn p

f(y)

|x–y|n–α

p dy,

whereΓn(α) = (1 –pα–n)/(1 –p–α) forα∈C,α= 0.

Lemma 2.3([14]) Letαbe a complex number with0 <Reα<n,and let1 <r<∞, 1 <q<

n/Reα,and0 < 1/r= 1/q–Reα/n.Suppose that bothωandνare nonnegative measurable functions such that

∞

j=γ

pjReα ν(Bj(a))

ω(Bγ(a))

=C<∞

for any a∈Qn

pandγ∈Z.Then the Riesz potential Ipαis bounded from GMq,νto GMr,ω.

3 Main results

In this section, we state the main results of the paper.

Theorem 3.1 Let1 <q<∞,and letΩ(pj_x_{) =}_Ω₍_x₎_{for all j}_∈_Z_,

|x|p=1Ω(x)dx= 0,and

sup

|y|p=1

∞

j=1

|x|p=1

Ωx+pjy –Ω(x)dx<∞.

Suppose that bothωandνare nonnegative measurable functions such that

∞

j=γ

νBj(a)/ω

Bγ(a) =C<∞ (3.1)

for anyγ∈Zand a∈Qn

p.Then the singular integral operators Tkare bounded from GMq,ν

to GMq,ωfor all k∈Z.Moreover,T(f) =limk→–∞Tk(f)exists in GMq,ω,and the operator

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Corollary 3.1 Let1 <q<∞,λ< 0,and letΩ∈L∞(Qn

p)be such thatΩ(pjx) =Ω(x)for all j∈Z,_|_x_|_p=1Ω(x)dx= 0,and

sup

|y|p=1

∞

j=1

|x|p=1

Then the operators Tkand T are bounded on the space Mq,λfor all k∈Z.

In fact, forλ< 0, takingω(B) =ν(B) =|B|λ

Hin Theorem3.1, we obtain Corollary3.1. If the

Morrey spaceMq,λ(Qn_p) is replaced by the central Morrey spaceCMq,λ(Qn_p) in Corollary3.1,

then the conclusion is that of Theorem 4.1 in [12].

Theorem 3.2 LetΩ∈L∞(Qn

p)be such thatΩ(pjx) =Ω(x)for all j∈Z,

|x|p=1Ω(x)dx= 0, and

sup

|y|p=1

∞

j=1

|x|p=1

Let0 <βi< 1for i= 1, 2, . . . ,m be such that0 <β=

m

i=1βi<n,and let1 <r<∞and

1 <q<n/βbe such that1/r= 1/q–β/n.Suppose that bi∈Λβi,i= 1, 2, . . . ,m,and bothω andνare nonnegative measurable functions such that

∞

j=γ

pjβνBj(a) /ω

Bγ(a) =C<∞ (3.2)

for anyγ ∈Zand a∈Qn

p.Then the commutators Tkbare bounded from GMq,νto GMr,ωfor

all k∈Z.Moreover,the commutator Tb₍_f_{) =}_lim

k→–∞Tkb(f)exists in the space of GMq,ω,

and Tb_{is bounded from GM}

q,νto GMq,ω.

Theorem 3.3 LetΩ∈L∞(Qn

|x|p=1Ω(x)dx= 0, and

sup

|y|p=1

∞

j=1

|x|p=1

Let1 <q,r,q1, . . . ,qm<∞be such that1/r= 1/q+ 1/q1+ 1/q2+· · ·+ 1/qm.Suppose thatω,ν, andνi(i= 1, 2, . . . ,m)are nonnegative measurable functions.Suppose that bi∈GCqi,νi(Q

n p), i= 1, 2, . . . ,m,and the functionsω,ν,andνi(i= 1, 2, . . . ,m)satisfy the following conditions:

(i) m_i=1νi(Bγ(a))ν(Bγ(a))/ω(Bγ(a)) =C<∞, (ii) ∞_j=γ+1

m

i=1νi(Bj(a))(j+ 1 –γ)mν(Bj(a))/ω(Bγ(a)) =C<∞

for anyγ ∈Zand a∈Qn

p.Then the commutators Tkbare bounded from GMq,νto GMr,ω

for all k∈Z.The commutator Tb₌_lim

k→–∞Tkb exists in the space of GMq,ω,and Tb is

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Corollary 3.2 LetΩ∈L∞(Qn

|x|p=1Ω(x)dx= 0, and

sup

|y|p=1

∞

j=1

|x|p=1

Let 1 <q,r,q1, . . . ,qm <∞ be such that 1/r= 1/q+ 1/q1 + 1/q2 +· · ·+ 1/qm. Let 0≤

λ1, . . . ,λm< 1/n, λ< –

m

i=1λi, andλ˜ =

m

i=1λi+λ.If bi∈BMOqi,λi(Qnp),then the com-mutators T_kband Tb_{are bounded from M}

q,λto M_r,λ˜ for all k∈Z.

Moreover, let 1 <r,q,q1<∞be such that 1/r= 1/q+ 1/q1. Let 0≤λ1< 1/n,λ< –λ1, and ˜

λ=λ1+λ. Ifb∈CBMOq1,λ1(Qnp), then from Corollary3.1it follows that the commutators Tb

k= [Tk,b] andTb= [T,b] are bounded fromCMq,λtoCMr,λ˜ for allk∈Z. These results

are those of Theorem 4.2 in [12].

4 Proof of Theorems3.1–3.3

Let us ﬁrst give the proof of Theorem3.1.

For any ﬁxedγ∈Zanda∈Qn_p, it is easy to see that 1

ω(Bγ(a))

1

|Bγ(a)|H

Bγ(a)

Tk(f)(x)qdx

1/q

≤ 1

ω(Bγ(a))

1

|Bγ(a)|H

Bγ(a)

Tk(f)(fχBγ(a))(x) q

dx

1/q

+ 1

ω(Bγ(a))

1

|Bγ(a)|H

Bγ(a)

Tk(fχBcγ(a))(x)

q dx

1/q

:=I+II, (4.1)

whereBcγ(a) is the complement toBγ(a) inQn_p.

Using Lemma2.2and (3.1), it follows that

I 1

ω(Bγ(a))

1

|Bγ(a)|1/q_H

Bγ(a)

f(x)qdx

1/q

= ν(Bγ(a))

ω(Bγ(a))

1

ν(Bγ(a))

1

|Bγ(a)|H

Bγ(a)

f(x)qdx

1/q

fGMq,ν. (4.2)

ForII, let us ﬁrst estimate|Tk(fχBc

γ(a))(x)|. Sincex∈Bγ(a) andΩ∈L∞(Qn_p), we have

Tk(fχBcγ(a))(x)= _|

y|p>pk

(fχBcγ(a))(x–y)

Ω(y)

|y|n p

dy

=

|x–z|p>pk(f

χBcγ(a))(z)

Ω(x–z)

|x–z|n p

dz

Bcγ(a)

|f(z)|

|x–z|n p

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∞

j=γ+1

Sj(a)

p–jnf(y)dy

≤

∞

j=γ+1 p–jn

Bj(a)

f(y)qdy

1/q

Bj(a) 1–1/q H

=fGMq,ν

∞

j=γ+1

νBj(a) . (4.3)

Thus from (3.1) and (4.3) it follows that

II = 1

ω(Bγ(a))

1

|Bγ(a)|H

Bγ(a)

Tk(fχBcγ(a))(x)

q dx

1/q

fGMq,ν

∞

j=γ+1

νBj(a)/ω

Bγ(a)

fGMq,ν. (4.4)

Combining the estimates of (4.1), (4.2), and (4.4), we have

1

ω(Bγ(a))

1

|Bγ(a)|H

Bγ(a)

Tk(f)(x) q

dx

1/q

fGMq,ν,

which means thatTkis bounded fromGMq,νtoGMq,ωfor allk∈Z.

Moreover, from Lemma2.2and the deﬁnition ofGMq,ω(Qnp) it is obvious thatT(f) = limk→–∞Tk(f) exists inGMq,ωand the operatorTis bounded fromGMq,νtoGMq,ω.

Proof of Theorem3.2 For anyx∈Qn_p, sinceΩ∈L∞(Qn_p) andbi∈Λβi,i= 1, 2, . . . ,m, it is

easy to see that

T_kbf(x)

≤

|y|p>pk m

i=1

bi(x) –bi(x–y)f(x–y)|

Ω(y)|

|y|n p

dy

Qn p

|f(z)|

|x–z|n–p β dz

I_pβ|f| (x).

Thus from Lemma2.3it is obvious that the commutatorsT_kbare bounded fromGMq,ν

toGMr,ωfor allk∈Z.

Moreover, from the deﬁnition ofGMq,ω(Qn_p) it is obvious thatTb(f) =limk→–∞Tkb(f)

exists in the space of GMq,ω, and the commutator Tb is bounded from GMq,ν to

GMq,ω.

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For any ﬁxedγ∈Zanda∈Qn

p, we writef0=fχBγ(a)andf∞=fχBcγ(a). Then

1

ω(Bγ(a))

1

|Bγ(a)|H

Bγ(a) _T(b1,b2)

k (f)(x) r

dx

1/r

≤ 1

ω(Bγ(a))

1

|Bγ(a)|H

Bγ(a)

b1(x) – (b1)Bγ(a) b2(x) – (b2)Bγ(a) Tk

f0 (x)rdx

1/r

+ 1

ω(Bγ(a))

1

|Bγ(a)|H

Bγ(a)

b1(x) – (b1)Bγ(a) Tk

b2– (b2)Bγ(a) f

0 ₍_x₎r_dx

1/r

+ 1

ω(Bγ(a))

1

|Bγ(a)|H

Bγ(a)

b1– (b1)Bγ(a) f

0 ₍_x₎r_dx

1/r

+ 1

ω(Bγ(a))

1

|Bγ(a)|H

Bγ(a) Tk

b1– (b1)Bγ(a) b2– (b2)Bγ(a) f

0 ₍_x₎r dx

1/r

+ 1

ω(Bγ(a))

×

1

|Bγ(a)|H

Bγ(a)

b1(x) – (b1)Bγ(a) b2(x) – (b2)Bγ(a) Tk

f∞ (x)rdx

1/r

+ 1

ω(Bγ(a))

×

1

|Bγ(a)|H

Bγ(a)

b2– (b2)Bγ(a) f

∞ ₍_x₎r dx

1/r

+ 1

ω(Bγ(a))

×

1

|Bγ(a)|H

Bγ(a)

b1– (b1)Bγ(a) f

∞ ₍_x₎r dx

1/r

+ 1

ω(Bγ(a))

1

|Bγ(a)|H

Bγ(a) Tk

b1– (b1)Bγ(a) b2– (b2)Bγ(a) f

∞ ₍_x₎r dx

1/r

=:E1+E2+E3+E4+E5+E6+E7+E8. (4.5)

We further estimate every part.

Since 1/r= 1/q+ 1/q1+ 1/q2, from Hölder’s inequality, Lemma2.2, and (i) it follows that

E1=

1

ω(Bγ(a))

×

1

|Bγ(a)|H

Bγ(a)

b1(x) – (b1)Bγ(a) b2(x) – (b2)Bγ(a) Tk

f0 (x)rdx

1/r

≤ 1

ω(Bγ(a))|Bγ(a)|1/r_H

2

i=1

Bγ(a)

bi(x) – (bi)Bγ(a)

qi dx

1/qi

× Bγ(a)

Tk

f0 (x)qdx

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ν1(Bγ(a))ν2(Bγ(a))

ω(Bγ(a))|Bγ(a)|1/q_H

2

i=1

biGC_qi,νi

Bγ(a)

f(x)qdx

1/q

≤ ν(Bγ(a))ν1(Bγ(a))ν2(Bγ(a))

ω(Bγ(a))

2

i=1

biGC_qi,νifGMq,ν

2

i=1

biGC_qi,ν_ifGMq,ν.

Let 1/q¯= 1/q+ 1/q2. Then 1/r= 1/q1+ 1/q¯. Thus, from Hölder’s inequality, Lemma2.2,

and (i) we obtain

E2=

1

ω(Bγ(a))

1

|Bγ(a)|H

Bγ(a)

b2– (b2)Bγ(a) f

0 ₍_x₎r_dx

1/r

≤ 1

Bγ(a)

b1(x) – (b1)Bγ(a)

q1

dx

1/q1

× Bγ(a)

Tk

b2– (b2)Bγ(a) f

0 ₍_x₎q¯ dx

1/q¯

1

Bγ(a)

b1(x) – (b1)Bγ(a)

q1

dx

1/q1

×

Bγ(a)

b2(x) – (b2)Bγ(a) f(x)

¯

q dx

1/q¯

≤ 1

ω(Bγ(a))|Bγ(a)|1/rH 2

i=1

Bγ(a)

qi dx

1/qi

Bγ(a)

f(x)qdx

1/q

ω(Bγ(a))

2

i=1

biGC_qi,ν_ifGMq,ν

2

i=1

biGC_qi,νifGMq,ν.

Similarly,

E3 2

i=1

ForE4, from Lemma2.2, Hölder’s inequality, and (i) we obtain

E4=

1

ω(Bγ(a))

1

|Bγ(a)|H

Bγ(a)

Tk

b1– (b1)Bγ(a) b2– (b2)Bγ(a) f0)(x) r

dx

1/r

1

Bγ(a)

b1(x) – (b1)Bγ(a) b2(x) – (b2)Bγ(a) f(x)

r dx

1/r

≤ 1

2

i=1

Bγ(a)

qi dx

1/qi

Bγ(a)

f(x)qdx

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ω(Bγ(a))

2

i=1

2

i=1

To estimateE5, we ﬁrst need to consider|Tk(f∞)(x)|. In fact, by (4.3) it is easy to see that

Tk

f∞ (x)fGMq,ν

∞

j=γ+1

νBj(a) . (4.6)

Therefore from Hölder’s inequality, (4.6), and (ii) we get

E5=

1

ω(Bγ(a))

×

1

|Bγ(a)|H

Bγ(a)

b1(x) – (b1)Bγ(a) b2(x) – (b2)Bγ(a) Tk

f∞ (x)rdx

1/r

≤ 1

2

i=1

Bγ(a)

bi(x) – (bi)Bγ(a) qi

dx

1/qi

× Bγ(a)

Tk

f∞ (x)f(x)qdx

1/q

∞

j=γ+1

ν(Bj(a))ν1(Bγ(a))ν2(Bγ(a))

ω(Bγ(a))

2

i=1

2

i=1

It is similar to estimate (4.3) forx∈Bγ(a). ByΩ∈L∞(Qnp) and (2.2) we can deduce that

_T_k

b2– (b2)Bγ(a) f

∞₎₍_x₎

=

|y|p>pk

b2(x–y) – (b2)Bγ(a) fχBcγ(a)(x–y)

Ω(y)

|y|n p

dy

≤

Bc

γ

b2(z) – (b2)Bγ(a)f(z)

|Ω(x–z)|

|x–z|n p

dz

Bcγ

|b2(z) – (b2)Bγ(a)||f(z)|

|x–z|n p

dz

∞

j=γ+1

Sj(a)

p–jnb2(z) – (b2)Bγ(a)f(y)dy

=

∞

j=γ+1

p–jnBj(a) 1–1/q–1/q2

H

Sj(a)

f(y)qdy

1/q

Sj(a)

b2(y) – (b2)Bγ(a)

q2

dy

(11)

≤ fGMq,ν

∞

j=γ+1

p–jnBj(a) 1–1/q2

H ν

Bj(a) Bj(a)

b2(y) – (b2)Bγ(a)

q2

dy

1/q2

b2GCq_2,ν2fGMq,ν

∞

j=γ+1

(j+ 1 –γ)νBj(a) ν2

Bj(a) . (4.7)

Let 1/q¯= 1/q+ 1/q2. Then 1/r= 1/q1+ 1/q¯. Thus from Hölder’s inequality, (4.7), and (ii)

it follows that

E6=

1

ω(Bγ(a))

1

|Bγ(a)|H

Bγ(a)

b2– (b2)Bγ(a) f

∞ ₍_x₎r dx

1/r

≤ 1

ω(Bγ(a))|Bγ(a)|1/rH

Bγ(a)

b1(x) – (b1)Bγ(a)

q1

dx

1/q1

× Bγ(a)

Tk

b2– (b2)Bγ(a) f

∞ ₍_x₎q¯

dx

1/q¯

≤ 2

i=1

biGC_qi,νifGMq,ν 1

ω(Bγ(a))

∞

j=γ+1

(j+ 1 –γ)νBj(a) ν2

Bj(a) ν1

Bγ(a)

2

i=1

Similarly estimatingE6, we obtain

E7 2

i=1

Moreover, sinceΩ∈L∞(Qn

p), by (2.2) it is easy to see that

Tk

b1– (b1)Bγ(a) b2– (b2)Bγ(a) f∞ (x)

=

|x–z|p>pk

b1(z) – (b1)Bγ(a) b2(z) – (b2)Bγ(a) fχBcγ(a)(z)

Ω(x–z)

|x–z|n p dz ≤ Bc γ

b1(z) – (b1)Bγ(a)b2(z) – (b2)Bγ(a)f(z)

|Ω(x–z)|

|x–z|n p

dz

∞

j=γ+1

Sj(a)

p–jnb1(z) – (b1)Bγ(a)b2(z) – (b2)Bγ(a)f(y)dy

=

∞

j=γ+1

p–jnBj(a)

1–1/q–1/q1–1/q2

H

Sj(a)

f(y)qdy

1/q

×

Sj(a)

b1(y) – (b1)Bγ(a)

q1 dy 1/q1 × Sj(a)

b2(y) – (b2)Bγ(a) q2 dy 1/q2 2 i=1

∞

j=γ+1

(j+ 1 –γ)2νBj(a)ν1

Bj(a) ν2

(12)

Therefore from (4.8) and (ii) we get that

E8=

1

ω(Bγ(a))

1

Bγ(a)|H

B

_T_k

b1– (b1)Bγ(a) b2– (b2)Bγ(a) f

∞ ₍_x₎r dx

1/r

≤ 2

i=1

biGC_qi,ν_ifGMq,ν 1

ω(Bγ(a))

∞

j=γ+1

(j+ 1 –γ)2νBj(a) ν1

Bj(a) ν2

Bj(a)

2

i=1

Combining (4.5) and the estimates ofE1,E2, . . . ,E8, we have

1

ω(Bγ(a))

1

|Bγ(a)|H

Bγ(a) T(b1,b2)

k (f)(x) r

dx

1/r ≤

2

i=1

biGC_qi,νifGMq,ν,

which means that the commutatorT(b1,b2)

k is bounded fromGMq,νtoGMr,ω.

Moreover, by Lemma2.2and the deﬁnition ofGMq,ω(Qn_p) it is obvious that the

commu-tatorTb₍_f_{) =}_lim

k→–∞Tkb(f) exists in the space ofGMq,ω, andTbis bounded fromGMq,ν

toGMq,ω.

Therefore the proof of Theorem3.3is complete.

5 Conclusion

In this paper, we established the boundedness of a class ofp-adic singular integral op-erators on thep-adic generalized Morrey spaces. We also considered the corresponding boundedness for the commutators generalized by thep-adic singular integral operators andp-adic Lipschitz functions orp-adic generalized Campanato functions.

Acknowledgements

The authors are grateful to the editor and referees for carefully reading the manuscript.

Funding

The work is supported by National Natural Science Foundation of China (No. 11601035).

Availability of data and materials

No data were used to support this study.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors have made equal contributions in this article. All authors read and approved the ﬁnal manuscript.

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional aﬃliations.

Received: 18 December 2018 Accepted: 25 February 2019

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